CHAP, ii] FORMULAS FOR NUMBER AND SUM OF DIVISORS. 53 sum of the aliquot parts of a, the sum of the aliquot parts of ap is bp+a+b. If b is the sum of the aliquot parts of a and if x is prime to a, the sum of the aliquot parts of axn is Descartes22 stated a result which may be expressed by the formula (1) <r(nm)=ff(n)<T(m) (n, m relatively prime), where <r(ri) is the sum of the divisors (including 1 and ri) of n. Here he solved n : <r(n) = 5 : 13. Thus n must be divisible by 5. Enter 5 in column A and <r(5) = 6 in column B. Then enter the factor 2 in column A and <r(2) =3 in column B. Having two threes in column B, we enter 9 in column A and <r(9) = 13 in B. Every 5 number except 13 in column B is in column A. Hence the 2 product 5-2-9 = 90 is a solution n. Next, to solve n : <r(ri) = 5 : 14, B 2-3 3 .13 we enter also 13 in column A and 14 in B, and obtain the solu- tion 9043. If n is a perfect number, 5n:or(5n) = 5: 12 and, if n?^6, 15n: <r(15n) = 5:16. Descartes23 stated that he possessed a general rule [illustrated above] for finding numbers having any given ratio to the sum of their aliquot parts. Fermat24 had treated the same problem. Replying to Mersenne's remark that the sum of the aliquot parts of 360 bears to 360 the ratio 9 to 4, Fermat25 noted that 2016 has the same property. John Wallis26 noted that Frenicle knew formula (1). Wallis27 knew the formula **1 P+l (2) ........ Thus these formulae were known before 1685, the date set by Peano,28 who attributed them to Wallis.52 G. W. Kraft29 noted that the method of Newton5 shows that the sum of the divisors of a product of distinct primes P,. . ., S is (P+l) . . .(S+ 1). He gave formula (1) and also (2), a formula which Cantor30 stated had probably not earlier been in print. To find a number the sum of whose divisors is a square, Kraft took PA, where P is a prune not dividing A. If cr(A)=a, then <r(PA) = (P+l)a will be the square of (P+l)B if P = 22"De la fagon de trouver le nombres de parties aliquotes in ratione data," manuscript Fonds-frangais, nouv. acquisitions, No. 3280, ff. 156-7, Bibliotheque Nationale, Paris. Published by C. Henry, Bull. Bibl. Storia Sc. Mat. e Fis., 12, 1879, 713-5. "Oeuvres, 2, p. 149, letter to Mersenne, May 27, 1638. "Oeuvres de Fermat, 2, top of p. 73, letter to Roberval, Sept. 22, 1636. "Oeuvres, 2, 179, letter to Mersenne, Feb. 20, 1639. MCommercium Epistolicum, letter 32, April 13, 1658; French transl. in Oeuvres de Fermat, 3, 553. "Commercium Epist., letter 23, March, 1658; Oeuvres de Fermat, 3, 515-7. "Formulaire Math., 3, Turin, 1901, 100-1. "Novi Comm. Ac. Petrop., 2, 1751, ad annum 1749, 100-109. »°Ge8chichte Math., 3, 595; ed. 2, 616.